Math Problem Statement
solve
Solution
Let's solve the problem step by step. We are given the equation of a curve in the form:
where and are integers. We also know that the curve has a stationary point at .
Part (a)
We are asked to show that and to find the value of .
Step 1: Differentiate the equation of the curve
To find the stationary points, we need to compute the derivative of with respect to . So, let's differentiate:
At a stationary point, the derivative is zero. We know the stationary point occurs at , so substitute into the derivative:
This confirms that .
Step 2: Find
We are also given that the curve passes through the point . Substitute , , and into the equation of the curve:
Thus, the value of is .
Part (b)
We are asked to state, giving a reason, the nature of the stationary point.
Step 1: Second derivative test
To determine the nature of the stationary point, we use the second derivative. Differentiate again with respect to :
Since , which is positive, the stationary point at is a minimum.
Part (c)
We are asked to find the equation of the normal to the curve at the point where .
Step 1: Find the gradient of the tangent at
The gradient of the tangent to the curve is given by the first derivative, . Substituting :
The gradient of the tangent is , so the gradient of the normal is the negative reciprocal of this:
Step 2: Find the coordinates of the point on the curve where
Substitute into the equation of the curve to find the -coordinate:
So, the point on the curve is .
Step 3: Equation of the normal
The equation of a line is given by:
where is the gradient of the normal, and is the point . Substituting the values:
Multiply through by 8 to eliminate the fraction:
Thus, the equation of the normal is:
Summary of answers:
- (a) ,
- (b) The stationary point is a minimum because , which is positive.
- (c) The equation of the normal is .
Do you have any questions or need further details?
Related Questions:
- How do you find the second derivative of more complex functions?
- What does the second derivative tell us about a curve's shape?
- Can there be stationary points that are neither maxima nor minima?
- How do we find the tangent to a curve at a given point?
- What role does the discriminant play in determining the nature of stationary points?
Tip: Always check if the second derivative is positive or negative at a stationary point to confirm whether it is a minimum or maximum.
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Math Problem Analysis
Mathematical Concepts
Calculus
Quadratic Functions
Derivatives
Stationary Points
Formulas
Derivative of y = 2x^2 + px + q is dy/dx = 4x + p
Second derivative: d^2y/dx^2 = 4
Equation of a line: y - y1 = m(x - x1)
Theorems
Second Derivative Test
Suitable Grade Level
Grades 10-12
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